The drift volume $D$ refers to the volume of fluid enclosed between the initial and final profiles of an initially flat marked sheet of fluid that deforms due to the passage of a body. Classic investigations of the drift volume in an inviscid fluid show that $D$ is comparable in magnitude to the volume of the translating body. The situation is fundamentally different for a viscous fluid. For instance, if the Reynolds number Re is zero, then $D$ diverges with the distance traveled by the body and is typically orders of magnitude larger than the body volume. The goal of the present paper is to quantify the drift volume when Re is finite. We interpret the drift volume as the flux through a stationary plane bounded by a stream tube, thus allowing $D$ to be computed without explicit reference to the trajectories of the marked fluid elements. We focus our analysis on a rigid sphere of radius $a$ that is steadily towed through an unbounded viscous fluid by an external force. The initial profile of the marked fluid is taken to be a disk of finite height $h$, and $h$ is furthermore assumed to be much larger than $a$. A two-term asymptotic expansion of $D$ for $h/a\to \infty$ is computed for $\text{Re}=0$ and compared to the analogous result for inviscid flow in order to illustrate the fundamental differences between these two cases. Next, a leading-order approximation to $D$ for a sphere translating at small Re is computed using Oseen's approximation to the flow. These results are extended to $\text{Re}\ge O\left(1\right)$ by taking advantage of the fact that, at distances greatly exceeding its radius, the flow around a sphere is described by the point force solution to Oseen's equations. Therefore, it is found universally that $D$ diverges as the distance traveled by the sphere becomes large at finite Re. However, the exact nature of this divergence critically depends upon the ratio of the distance traveled by the sphere to the radius of the initially marked disk of fluid ($h$).

Multiscale directional statistics are investigated in two-dimensional incompressible turbulence. It is shown that the short-time behavior of the mean angle of directional change of fluid particles is linearly dependent on the time lag and that no inertial range behavior is observed in the directional change associated with the enstrophy-cascade range. In simulations of the inverse-cascade range, the directional change shows a power law behavior at inertial range time scales. By comparing the directional change in space-periodic and wall-bounded flow, it is shown that the probability density function of the directional change at long times carries the signature of the confinement. The geometrical origin of this effect is validated by Monte Carlo simulations. The same effect is also observed in the directional statistics computed from the trajectories of football players (soccer players in American English).

Partial coalescence of microdroplets is of interest for a number of microfluidic applications where a controlled fluid transfer from one droplet to another is required for mixing, dispensing, and metering of chemical and biological fluids. We report a phenomenon of oscillatory coalescence of water droplets situated in an alternating electric field. The oscillatory coalescence exists in a range of electric capillary numbers and fluid conductivities and proceeds through a finite number of cycles. Each cycle includes attractive and repulsion stages and results in a partial fluid transfer through a liquid bridge formed between droplets during the repulsion stage. We propose an energy model to describe the phenomenon and define its limit of existence.

High-speed, time-resolved particle image velocimetry with a pulse-burst laser was used to measure the gas-phase velocity upstream and downstream of a shock wave–particle curtain interaction at three shock Mach numbers (1.22, 1.40, and 1.45) at a repetition rate of 37.5 kHz. The particle curtain was formed from free-falling soda-lime particles resulting in volume fractions of 9% or 23% at mid-height, depending on particle diameter (106–125 and 300–355 $\mu \mathrm{m}$, respectively). Following impingement by a shock wave, a pressure difference was created between the upstream and downstream sides of the curtain, which accelerated flow through the curtain. Jetting of flow through the curtain was observed downstream once deformation of the curtain began, demonstrating a long-term unsteady effect. Using a control volume approach, the unsteady drag on the curtain was estimated from velocity and pressure data. The drag imposed on the curtain has a strong volume fraction dependence with a prolonged unsteadiness following initial shock impingement. In addition, the data suggest that the resulting pressure difference following the propagation of the reflected and transmitted shock waves is the primary component to curtain drag.

We consider the dynamics of a small spherical particle driven through an unbounded viscoelastic shear flow by an external force. We give analytical solutions to both the mobility problem (the velocity of a forced particle) and the resistance problem (the force on a fixed particle), valid to second order in the dimensionless Deborah and Weissenberg numbers, which represent the elastic relaxation time of the fluid relative to the rate of translation and the imposed shear rate. We find a shear-induced lift at $O\left(\mathrm{Wi}\right)$, a modified drag at $O\left({\mathrm{De}}^2\right)$ and $O\left({\mathrm{Wi}}^2\right)$, and a second lift that is orthogonal to the first, at $O\left({\mathrm{Wi}}^2\right)$. The relative importance of these effects depends strongly on the orientation of the forcing relative to the shear. We discuss how these forces affect the terminal settling velocity in an inclined shear flow. We also describe a basis set of symmetric Cartesian tensors and demonstrate how they enable general tensorial perturbation calculations such as the present theory. In particular, this scheme allows us to write down a solution to the inhomogeneous Stokes equations, required by the perturbation expansion, by a sequence of algebraic manipulations well suited to computer implementation.

The dissolution of a gas bubble in a confined geometry is a problem of interest in technological applications such as microfluidics or carbon sequestration, as well as in many natural flows of interest in geophysics. While the dissolution of spherical or sessile bubbles has received considerable attention in the literature, the case of a two-dimensional bubble in a Hele-Shaw cell, which constitutes perhaps the simplest possible confined configuration, has been comparatively less studied. Here, we use planar laser-induced fluorescence to experimentally investigate the diffusion-driven transport of dissolved ${\mathrm{CO}}_2$ that propagates from a cylindrical mm-sized bubble in air-saturated water confined in a horizontal Hele-Shaw cell. We observe that the radial trajectory of an isoconcentration front, $r_f\left(t\right)$, evolves in time as approximately $r_f-R_0\propto \sqrt{t}$, where $R_0$ denotes the initial bubble radius. We then characterize the unsteady ${\mathrm{CO}}_2$ concentration field via two simple analytical models, which are then validated against a numerical simulation. The first model treats the bubble as an instantaneous line source of ${\mathrm{CO}}_2$, whereas the second assumes a constant interfacial concentration. Finally, we provide an analogous Epstein-Plesset equation with the intent of predicting the dissolution rate of a cylindrical bubble.

We present a theoretical study of the unsteady thin-film dynamics of a long bubble of negligible viscosity that advances at a uniform speed in a cylindrical capillary tube. The bubble displaces a viscous nonwetting fluid, creating a thin film between its interface and the tube walls. The film is considered thin enough that intermolecular forces in the form of van der Waals attractions are significant and thin-film rupture is possible. In the case of negligible intermolecular forces, a steady-state solution exits where a film of uniform thickness is deposited in the annular region between the bubble interface and the tube walls. However, once intermolecular interactions are important, the interface is perturbed out of its steady state and either (i) the perturbation grows sufficiently before reaching the rear meniscus of the bubble such that rupture occurs or (ii) the perturbation remains small due to weak intermolecular forces until it leaves the bubble interface through the rear meniscus. We obtain, both numerically and asymptotically, the time scale over which rupture occurs and thus we find a critical capillary number, depending on the bubble length and the strength of the intermolecular forces, below which the film is predicted to rupture.

To draw a precise picture of the hierarchy of coherent vortices in spatially periodic turbulence at high Reynolds numbers and to understand its generation mechanism, we conduct direct numerical simulations of turbulence in a periodic cube. By objectively identifying the axes of vortex tubes at various length scales in the inertial range, we quantitatively show that the sustained turbulence consists of a hierarchy of antiparallel pairs of vortex tubes. These vortex tubes are created by being stretched in strain fields around 2–8 times larger vortices, whereas they are weakened by strain around half-scale vortices. Temporal tracking of identified vortex tubes shows that they tend to form antiparallel pairs from the moment they are created. We examine three different external forces to show that the hierarchical structure of antiparallel vortex pairs is robust and its statistical features in inertial length scales are independent of the force. The turbulence that is sustained by steady forces is quasiperiodic in time and its significant temporal fluctuations are caused by successive creation of strong vortex tubes within the hierarchy.

We investigate the scaling of the velocity structure function tensor $D_{ij}(\mathbf{r},z)$ in high Reynolds number wall-bounded turbulent flows, within the framework provided by the Townsend attached eddy hypothesis. Here $i,j=1,2,3$ denote velocity components in the three Cartesian directions, and $\mathbf{r}$ is a general spatial displacement vector. We consider spatial homogeneous conditions in wall-parallel planes and dependence on wall-normal distance is denoted by $z$. At small scales ($r=\left|\mathbf{r}\right|\ll z$) where turbulence approaches local isotropy, $D_{ij}(\mathbf{r},z)$ can be fully characterized as a function of $r$ and the height-dependent dissipation rate $\epsilon \left(z\right)$, using the classical Kolmogorov scalings. At larger distances in the logarithmic range, existing previous studies have focused mostly on the scaling of $D_{ij}$ for $\mathbf{r}$ in the streamwise direction and for the streamwise velocity component ($i=j=1$) only. No complete description is available for $D_{ij}(\mathbf{r},z)$ for all $i,j$, and $\mathbf{r}$ directions. In this paper we show that the hierarchical random additive process model for turbulent fluctuations in the logarithmic range (a model based on the Townsend's attached eddy hypothesis) may be used to make new predictions on the scaling of $D_{ij}(\mathbf{r},z)$ for all velocity components and in all two-point displacement directions. Some of the generalized scaling relations of $D_{ij}(\mathbf{r},z)$ in the logarithmic region are then compared to available data. Nevertheless, a number of predictions cannot yet be tested in detail, due to a lack of simultaneous two-point measurements with arbitrary cross-plane displacements, calling for further experiments to be conducted at high Reynolds numbers.

This computational study analyzes the effect of variation of the radius of gyration ($r_g$), expressed as the Rossby number $\text{Ro}=r_g/C$, with $C$ the chord, on the aerodynamics of a rotating wing at a Reynolds number of 1400. The wing is represented as an aspect-ratio-unity rectangular flat plate aligned at $45{}^{\circ }$. This plate is accelerated near impulsively to a constant rotational velocity and the flow is allowed to develop. Flow structures are analyzed and force coefficients evaluated. Trends in velocity field degradation with increasing Ro are consistent with previous experimental studies. At low Ro the flow structure generated initially is mostly retained with a strong laminar leading-edge vortex (LEV) and tip vortex (TV). As both Ro and travel distance increase, the flow structure degrades such that at high Ro it begins to resemble that of a translating wing. Additionally, the present study has shown the following. (i) At low Ro the LEV and TV structure is laminar and steady; as Ro increases this structure breaks down, and the location at which it breaks down shifts closer to the wing root. (ii) For moderate Ro of 1.4 and higher, the LEV is no longer steady but enters a shedding regime fed by the leading-edge shear layer. (iii) At the lowest Ro of 0.7 the lift force rises during start-up and then stabilizes, consistent with the flow structure being retained, while for higher Ro a force peak occurs after the initial acceleration is complete, followed by a reduction in lift which appears to correspond to shedding of excess leading-edge vorticity generated during start-up. (iv) All rotating wings produced greater lift than a translating wing, this increase varied from $\sim 65\%$ at the lowest $\text{Ro}=0.7$ down to $\sim 5\%$ for the highest Ro examined of 9.1.

Sound radiation from a subsonic turbulent jet is examined after a hypothetical removal of the near-field coherent structures in the axisymmetric component of the velocity fluctuations. With the help of a well-validated database of large-eddy simulation, the near-field coherent structures are extracted using a discrete wavelet transform (DWT), and their spatial structures are examined using a proper orthogonal decomposition (POD). The acoustic far field is calculated using Lighthill's acoustic analogy. It is shown that the coherent part extracted by DWT accounts for most of the fluctuation energy in the axisymmetric component of axial velocity, whereas the incoherent part, assumed to have a Gaussian probability distribution, has little energy. After the coherent part is removed, the axisymmetric component of the sound is found to be significantly reduced, around 7 dB in the overall sound pressure level at 30 deg with respect to the jet axis. The reduction is mostly at low Strouhal numbers ($\text{St}<0.4$, based on the speed of sound and the nozzle exit diameter). The first few POD modes of the near-field coherent part, which capture most of the fluctuation energy, are found to be characterized by large-scale wavy structures. After these POD modes are removed, the axisymmetric component of the sound pressure level is also reduced considerably, by around 5 $\mathrm{dB}/\text{St}$ at $\text{St}=0.2$. The results suggest that there is a causal link between the axisymmetric components of the near-field hydrodynamic fluctuations and far-field low-angle jet noise, although the axisymmetric mode constitutes only a small proportion of total fluctuation energy. It is also suggested that not only the large-scale wavy structures in low POD modes but also the smaller scale structures in higher POD modes need to be included for jet noise modeling, because they are both shown to be efficient at sound radiation.

Plasmonically absorbing nanofluids exhibit light-induced electrokinetics. We measure an electrical response to the light-induced Rayleigh-Bénard-Marangoni convective instabilities in gold-polyvinylpyrrolidone (PVP) nanoparticles (NPs) suspended in isopropanol and water. Microampere current oscillations are measured and attributed to the presence of the Au-PVP NPs with negative zeta potential, in correspondence with the accompanying thermal lens oscillations and a nanofluid thermoelectric effect. The measured electrical oscillations represent an electrohydrodynamic stability driven by light, one among many that should be observed with plasmonic nanoparticles in liquids.

Detonation propagation in a compressible medium wherein the energy release has been made spatially inhomogeneous is examined via numerical simulation. The inhomogeneity is introduced via step functions in the reaction progress variable, with the local value of energy release correspondingly increased so as to maintain the same average energy density in the medium and thus a constant Chapman-Jouguet (CJ) detonation velocity. A one-step Arrhenius rate governs the rate of energy release in the reactive zones. The resulting dynamics of a detonation propagating in such systems with one-dimensional layers and two-dimensional squares are simulated using a Godunov-type finite-volume scheme. The resulting wave dynamics are analyzed by computing the average wave velocity and one-dimensional averaged wave structure. In the case of sufficiently inhomogeneous media wherein the spacing between reactive zones is greater than the inherent reaction zone length, average wave speeds significantly greater than the corresponding CJ speed of the homogenized medium are obtained. If the shock transit time between reactive zones is less than the reaction time scale, then the classical CJ detonation velocity is recovered. The spatiotemporal averaged structure of the waves in these systems is analyzed via a Favre-averaging technique, with terms associated with the thermal and mechanical fluctuations being explicitly computed. The analysis of the averaged wave structure identifies the super-CJ detonations as weak detonations owing to the existence of mechanical nonequilibrium at the effective sonic point embedded within the wave structure. The correspondence of the super-CJ behavior identified in this study with real detonation phenomena that may be observed in experiments is discussed.

Numerical simulations based on DEM-CFD were conducted to study the behavior of gas-solid flows of cohesive particles under various values of particle friction and restitution coefficient, which dictate the energy dissipation in the tangential and normal directions of particle relative motion. Fluidized beds and riser flows were selected as typical systems for dense and dilute flows, respectively. Based on the defluidization curves and agglomerate properties in respective systems, a reverse dependence on friction and restitution coefficient was identified: defluidization curves were dominated by friction while agglomerates in riser flow were governed by the restitution coefficient. The reverse sensitivity is ascribed to the difference in particle interactions for the two systems. In the fluidized bed, particles primarily interact via enduring multiple-particle contacts, in which the dynamics in the tangential directions dominates. In riser flows, the instantaneous binary collisions are more common and the relative motion of particles in the normal direction becomes important. A nonmonotonic response of defluidization curves to varying sliding friction was observed, which is explained by the competing effects of increased sliding and enhanced spin of particles on bed porosity. This study highlights the importance of correct experimental measurement of solid properties for numerical simulations. The results are also useful for driving the development of continuum modeling of gas-solid flows of cohesive particles.

We use high-resolution direct numerical simulations to study the anisotropic contents of a turbulent, statistically homogeneous flow with random transitions among multiple energy containing states. We decompose the velocity correlation functions on different sectors of the three-dimensional group of rotations, $\text{SO}\left(3\right)$, using a high-precision quadrature. Scaling properties of anisotropic components of longitudinal and transverse velocity fluctuations are accurately measured at changing Reynolds numbers. We show that independently of the anisotropic content of the energy containing eddies, small-scale turbulent fluctuations recover isotropy and universality faster than previously reported in experimental and numerical studies. The discrepancies are ascribed to the presence of highly anisotropic contributions that have either been neglected or measured with less accuracy in the foregoing works. Furthermore, the anomalous anisotropic scaling exponents are devoid of any sign of saturation with increasing order. Our study paves the way to systematically assess persistence of anisotropy in high-Reynolds-number flows.

We consider the one-dimensional Majda-McLaughlin-Tabak (MMT) model that describes the interactions of nonlinear and dispersive waves. We perform a detailed numerical study of the direct energy cascade. Our numerical experiments show the following. (i) In the limit of small nonlinearity the spectral slope observed in the statistical steady regime corresponds to the one predicted by the weak-wave-turbulence (WWT) theory. (ii) As the nonlinearity is increased, the WWT theory breaks down and deviations from its predictions are observed. (iii) It is shown that such departures from the WWT theoretical predictions are accompanied by the phenomenon of intermittency, typical of three-dimensional fluid turbulence. Our results clarify the role played by the wave-turbulence theory in the statistical description of nonlinear-dispersive waves described by the MMT model.

Scaling and intermittency in compressible isotropic turbulence at the turbulent Mach number $M_t$ ranging from 0.5 to 1.0 are studied by using numerical simulations with solenoidal forcing. Linear relations between the structure functions of the compressible velocity component and those of thermodynamic variables are modeled based on the shock jump conditions and are verified by numerical simulations. At a turbulent Mach number around 1.0, the relative scaling exponent of the structure functions saturates with an increase of the order. After proper normalization, the tails of the probability density functions (PDFs) of the increments of the compressible velocity component and thermodynamic variables overlap one another for different separations. Moreover, we study the conditional PDFs of the increments with respect to the shocklet. Linear relations between the tails of unconditional PDFs and conditional PDFs are established. The shocklet plays an important role in the determination of the PDF tails. The compressible velocity increment is decomposed into a negative component and a positive component. The negative component of the compressible velocity increment exhibits a scaling behavior with the saturation of the scaling exponent at high orders, which is similar to the Burgers turbulence, while the positive component of the compressible velocity increment exhibits a power-law scaling behavior, which is similar to the incompressible turbulence.

Three-dimensional (3D) turbulence is characterized by a dual forward cascade of both kinetic energy and helicity, a second inviscid flow invariant besides energy, from the integral scale of motion to the viscous dissipative scale. In helical flows, however, such as strongly rotating flows with broken mirror symmetry, an inverse (reversed) energy cascade can be observed analogous to that of two-dimensional turbulence (2D) where enstrophy, a second positive-definite flow invariant, unlike helicity in 3D, effectively blocks the forward cascade of energy. In the spectral-helical decomposition of the Navier-Stokes equation, it has previously been shown that a subset of three-wave (triad) interactions conserve helicity in 3D in a fashion similar to enstrophy in 2D, thus leading to a 2D-like inverse energy cascade in 3D. In this work, we show, both theoretically and numerically, that an additional subset of interactions exist, conserving a new pseudo-invariant in addition to energy and helicity, which contributes either to a forward or an inverse energy cascade depending on the specific triad interaction geometry.

Multicomponent turbulence measurements in rough-wall boundary layers are presented and compared to smooth-wall data over a large friction Reynolds number range (${\delta }^{+}$). The rough-wall experiments used the same continuous sandpaper sheet as in the study of Squire et al. [J. Fluid Mech. 795, 210 (2016)]. To the authors' knowledge, the present measurements are unique in that they cover nearly an order of magnitude in Reynolds number (${\delta }^{+}\simeq 2800–17400$), while spanning the transitionally to fully rough regimes (equivalent sand-grain-roughness range, $k_s^{+}\simeq 37-98$), and in doing so also maintain very good spatial resolution. Distinct from previous studies, the inner-normalized wall-normal velocity variances, $\overline{w^2}$, exhibit clear dependencies on both $k_s^{+}$ and ${\delta }^{+}$ well into the wake region of the boundary layer, and only for fully rough flows does the outer portion of the profile agree with that in a comparable ${\delta }^{+}$ smooth-wall flow. Consistent with the mean dynamical constraints, the inner-normalized Reynolds shear stress profiles in the rough-wall flows are qualitatively similar to their smooth-wall counterparts. Quantitatively, however, at matched Reynolds numbers the peaks in the rough-wall Reynolds shear stress profiles are uniformly located at greater inner-normalized wall-normal positions. The Reynolds stress correlation coefficient, $R_{uw}$, is also greater in rough-wall flows at a matched Reynolds number. As in smooth-wall flows, $R_{uw}$ decreases with Reynolds number, but at different rates depending on the roughness condition. Despite the clear variations in the $R_{uw}$ profiles with roughness, inertial layer $u, w$ cospectra evidence invariance with $k_s^{+}$ when normalized with the distance from the wall. Comparison of the normalized contributions to the Reynolds stress from the second quadrant (Q2) and fourth quadrant (Q4) exhibit noticeable differences between the smooth- and rough-wall flows. The overall time fraction spent in each quadrant is, however, shown to be nearly fixed for all of the flow conditions investigated. The data indicate that at fixed ${\delta }^{+}$ both Q2 and Q4 events exhibit a sensitivity to $k_s^{+}$. The present results are discussed relative to the combined influences of roughness and Reynolds number on the scaling behaviors of boundary layers.